The Chemistry Maths Book, Second Edition

13.5 The Hermite equation 381

The integrand is an even function of xin the interval− 11 ≤ 1 x 1 ≤ 11. Therefore

(Section 5.3, equation (5.25))

(ii)P

2

2

(x) 1 = 1 3(1 1 − 1 x

2

), P

3

2

(x) 1 = 115 x(1 1 − 1 x

2

)

The integrand is an odd function of xin the interval − 11 ≤ 1 x 1 ≤ 11 , and I 1 = 10

(Section 5.3, equation (5.26))

0 Exercise 20

13.5 The Hermite equation

The Hermite equation is

y′′ 1 − 12 xy′ 1 + 12 ny 1 = 10 (13.30)

where nis a real number. The equation arises in the solution of the Schrödinger

equation for the harmonic oscillator. It is solved by the power-series method to give a

general solution

y(x) 1 = 1 a

0

y

1

(x) 1 + 1 a

1

y

2

(x)

in which the particular solutiony

1

contains only even powers of xandy

2

only odd

powers of x. As in the case of the Legendre equation,y

1

reduces to a polynomial of

degree nwhen (positive) nis an even integer, and similarly fory

2

when nis an odd

integer. These polynomials are called Hermite polynomialsH

n

(x), and they are those

solutions of the Hermite equation that are of interest in the physical sciences:

(13.31)

forn 1 = 1 0, 1, 2, 3, =The first few of these polynomials are

H

0

(x) 1 = 11 H

1

(x) 1 = 12 x

H

2

(x) 1 = 14 x

2

1 − 12 H

3

(x) 1 = 18 x

3

1 − 112 x (13.32)

Hx x

nn

x

nn n n

n

nn

() ( )

()

()

()( )(

=−

−

!

• 
  

−− −

−

2

1

1

2

123

2

))

()

2

2

4

!

−

−

x

n



IPxPxdx xxdx==−

−

+

−

+

ZZ

1

1

2

2

3

2

1

1

22

() () 45 ( ) 1

=−+ −













32 0=

53

0

1

xxx

I=− −=−+−xxdx xxdx

++

31 51 3 561

0

1

22

0

1

42

ZZ()( ) ( )